Showing papers on "Mixture theory published in 2006"

Hygro‐thermo‐chemo‐mechanical modelling of concrete at early ages and beyond. Part I: hydration and hygro‐thermal phenomena

Abstract: Part 1 of the paper presents a new numerical model of hygro-thermal and hydration phenomena in concrete at early ages and beyond. This is a solidification-type model where all changes of material properties are expressed as functions of hydration degree, and neither as maturity nor as equivalent hydration period as in maturity-type models. A mechanistic approach has been used to obtain the governing equations, by means of an averaging theory of Hassanizadeh and Gray, also called hybrid mixture theory. The developments start at the micro-scale and balance equations for phases and interfaces are introduced at this level and then averaged for obtaining macroscopic balance equations. Constitutive laws are directly introduced at macroscopic level. The final equations, mass (water species and dry air), energy and momentum balance equations, have been written in terms of the chosen primary variables: gas pressure, capillary pressure, temperature and displacements. An evolution equation for the internal variable, hydration degree, describes hydration rate as a function of chemical affinity, considering in addition to the existing models, an effect of the relative humidity on the process. The model takes into account full coupling between hygral, thermal and chemical phenomena, as well as changes of concrete properties caused by hydration process, i.e. porosity, density, permeability, and strength properties. Phase changes and chemical phenomena, as well as the related heat and mass sources are considered. Two examples showing possibilities of the model for analysis of autogenous self-heating and self-desiccation phenomena, as well as influence of the ambient relative humidity and the concrete element dimensions upon hygro-thermal performance and shrinkage of the elements, are presented and discussed. Copyright © 2006 John Wiley & Sons, Ltd.

Bayesian feature and model selection for Gaussian mixture models

Abstract: We present a Bayesian method for mixture model training that simultaneously treats the feature selection and the model selection problem. The method is based on the integration of a mixture model formulation that takes into account the saliency of the features and a Bayesian approach to mixture learning that can be used to estimate the number of mixture components. The proposed learning algorithm follows the variational framework and can simultaneously optimize over the number of components, the saliency of the features, and the parameters of the mixture model. Experimental results using high-dimensional artificial and real data illustrate the effectiveness of the method.

Mathematical modelling of engineered tissue growth using a multiphase porous flow mixture theory

Abstract: This paper outlines the framework of a porous flow mixture theory for the mathematical modelling of in vitro tissue growth, and gives an application of this theory to an aspect of tissue engineering. The problem is formulated as a set of partial differential equations governing the space and time dependence of the amounts of each component of the tissue (phase), together with the physical stresses in each component. The theory requires constitutive relations to specify the material properties of each phase, and also requires relations to specify the stresses developed due to mechanical interactions, both within each phase and between different phases. An application of the theory is given to the study of the mobility and aggregation of a population of cells seeded into an artificial polymeric scaffold. Stability analysis techniques show that the interplay of the forces between the tissue constituents results in two different regimes: either the cells form aggregates or disperse through the scaffold.

A numerical method for the evaluation of non-linear transient moisture flow in cellulosic materials

Abstract: A numerical method for the transient moisture flow in porous cellulosic materials like paper and wood is presented. The derivation of the model is based on mass conservation for a mixture containing a vapour phase and an adsorbed water phase embedded in a porous solid material. The principle of virtual moisture concentrations in conjunction with a consistent linearization procedure is used to produce the iterative finite element equations. A monolithic solution strategy is chosen in order to solve the coupled non-symmetric equation system. A model for the development of higher order sorption hysteresis is also developed. The model is capable of describing cyclic hardening as well as cyclic softening of the equilibrium water concentration. The model is verified by comparison with the measured response to natural variations in temperature and humidity. A close agreement of the simulated results to measured data is found. Copyright (c) 2005 John Wiley & Sons, Ltd. (Less)

An Elasto-Viscoplastic Model and Multiphase Coupled FE Analysis for Unsaturated Soil

Abstract: Rate sensitivity is an important characteristic of geomaterials for both saturated and unsaturated soils. However, many constitutive models for unsaturated soil have been constructed within the framework of the rate independent theory. The present study addresses an elasto-viscoplastic constitutive model which considers the effect of suction for unsaturated clayey soil and a soil-water-air three-phase coupled analysis using the elasto-viscoplastic model. The proposed constitutive model adopts the average skeleton stress for the effective stress from the viewpoint of the mixture theory. Hence, it has become possible to construct a model for unsaturated soil starting with a model for saturated soil by substituting the average skeleton stress for the effective stress and introducing the suction effect into the constitutive model. Furthermore, the collapse behavior, which is brought about by a decrease in suction, is described by the shrinkage of the overconsolidation boundary surface, the static yield surface, and the viscoplastic potential surface. A numerical analysis for multiphase materials is conducted within the framework of a continuum mechanics approach through the use of the theory of porous media. The theory is a generalization of Biot's two-phase mixture theory for saturated soil. A soil-water-air three-phase coupled finite element method is developed in the present study using the governing equations for multiphase soil based on the non-linear finite deformation theory. The average skeleton stress is defined as the difference between the total stress and the average pressure of the two fluids and is used in the proposed elasto-viscoplastic constitutive model. A van Genuchten (1980) type of equation is employed as the constitutive equation between the liquid saturation and the suction pressure. Numerical simulations of unexhausted-undrained compression with different strain rates are conducted under plane strain conditions, and the applicability of the proposed method is evaluated with respect to strain localization and the effect of suction.

Tracking facial features using mixture of point distribution models

Abstract: We present a generic framework to track shapes across large variations by learning non-linear shape manifold as overlapping, piecewise linear subspaces. We use landmark based shape analysis to train a Gaussian mixture model over the aligned shapes and learn a Point Distribution Model(PDM) for each of the mixture components. The target shape is searched by first maximizing the mixture probability density for the local feature intensity profiles along the normal followed by constraining the global shape using the most probable PDM cluster. The feature shapes are robustly tracked across multiple frames by dynamically switching between the PDMs. Our contribution is to apply ASM to the task of tracking shapes involving wide aspect changes and generic movements. This is achieved by incorporating shape priors that are learned over non-linear shape space and using them to learn the plausible shape space. We demonstrate the results on tracking facial features and provide several empirical results to validate our approach. Our framework runs close to real time at 25 frames per second and can be extended to predict pose angles using Mixture of Experts.

Single-Channel mixture decomposition using bayesian harmonic models

Abstract: We consider the source separation problem for single-channel music signals. After a brief review of existing methods, we focus on decomposing a mixture into components made of harmonic sinusoidal partials. We address this problem in the Bayesian framework by building a probabilistic model of the mixture combining generic priors for harmonicity, spectral envelope, note duration and continuity. Experiments suggest that the derived blind decomposition method leads to better separation results than nonnegative matrix factorization for certain mixtures.

A Multi-Phase Coupled FE Analysis Using an Elasto-Viscoplastic Model for Unsaturated Soil

Abstract: The present study addresses an elasto-viscoplastic constitutive model which considers the effect of suction in unsaturated clayey soil and a soil-water-air three-phase coupled analysis using the elasto-viscoplastic model. The proposed constitutive model adopts the average skeleton stress for the effective stress from mixture theory. Hence, it has become possible to construct a model for unsaturated soils starting with a model for a saturated soil by substituting the average skeleton stress for the effective stress and introducing for the suction effect into the constitutive model. Furthermore, the collapse behavior, which is brought about by a decrease in suction can be described by the shrinkage of the overconsolidation boundary surface, the static yield surface, and the viscoplastic potential surface. A numerical analysis for multiphase materials is conducted within the framework of a continuum mechanics approach through the use of the theory of porous media. The theory is a generalization of Biot's two-phase mixture theory for saturated soil. A soil-water-air three-phase coupled finite element method has been developed in the present study using the governing equations for multi-phase soil based on the non-linear finite deformation theory. The average skeleton stress is defined as the difference between the total stress and the average pressure of the two fluids and is used in the proposed elasto-viscoplastic constitutive model. A van Genuchten (1980) type of equation is employed as the constitutive equation between the liquid saturation and the suction pressure. Numerical simulations of unexhausted-undrained compression are conducted under plane strain conditions, and the applicability of the proposed method is evaluated with respect to strain localization and the effect of suction.

Thermomechanics of Porous Media - I: Thermohydraulic Model for Compacted Bentonite

Abstract: A general thermomechanical model is derived for a mixture. The model describes the behavior of the mixture via proper choices of free energy and dissipation function. A model for any combination of the mixture constituents can be reduced from the general model. The theory is applied to a thermohydraulic model for a mixture of compacted bentonite, liquid water, vapor, and air with the assumption of rigid skeleton and constant uniform porosity. The free energy of the system is chosen to take into account the individual nondissipative behaviors of the constituents and their mutual interactions, namely, adsorption and mixing of the gaseous constituents. The choices for the interaction terms are based on the equilibrium conditions for the water species in different combinations of the constituents. The resulting thermodynamically consistent macroscopic model is fitted to a suction experiment and applied to a simple one-dimensional thermohydraulic simulation of the bentonite buffer of the Febex in situ test. The results calculated with finite element method are successfully compared to measurements.

Color image segmentation through unsupervised gaussian mixture models

Abstract: In this paper we introduce a novelty EM based algorithm for Gaussian Mixture Models with an unknown number of components. Although the EM (Expectation-Maximization) algorithm yields the maximum likelihood solution it has many problems: (i) it requires a careful initialization of the parameters; (ii) the optimal number of kernels in the mixture may be unknown beforehand. We propose a criterion based on the entropy of the pdf (probability density function) associated to each kernel to measure the quality of a given mixture model, and a modification of the classical EM algorithm to find the optimal number of kernels in the mixture. We apply our algorithm to the unsupervised color image segmentation problem.

Pre-stressed and reinforced hollow cylindrical mixture of non-linearly elastic solid and ideal fluid subjected to combined deformations: A study within the context of the theory of interacting continua

Abstract: The theoretical result dealing with the saturation boundary condition, first investigated by [On boundary conditions for a certain class of problems in mixture theory, Int. J. Eng. Sci. 24 (1986) 1453–1463] , has been recently extended by [On the saturation boundary condition within the context of the theory of interacting continua containing a certain distribution of fibers, Int. J. Eng. Sci. 41 (2003) 2273–2280] to the case of fibers-reinforced solid-ideal fluid mixture. Taking advantage of this new result, the problem of a hollow cylindrical mixture subjected to combined deformations, previously treated by Gandhi et al. [Some non-linear diffusion problems within the context of interacting continua, Int. J. Eng. Sci. 25 (1987) 1441–1457], is revisited in this contribution and improved by introducing the presence of fibers and residual stress (or pre-stress) characterized by the opening angle of the sector-like cross-section.

Solid-fluid interaction analysis by using a multi-material Eulerian finite element method

Abstract: A multi-material Eulerian finite element method for simulating solid-fluid interaction analyses is proposed in this paper. Although a conventional solid-fluid interaction code models solid and fluid in a Lagrangian and an Eulerian formulation respectively, the present computational framework bases on just the Eulerian formulation for solid and fluid. A mixture theory is used for handling multi-material (solid and fluid) elements. Solid and fluid element stresses are then mixed by considering their density function.

Two entropy-based methods for learning unsupervised gaussian mixture models

Abstract: In this paper we address the problem of estimating the parameters of a Gaussian mixture model. Although the EM (Expectation-Maximization) algorithm yields the maximum-likelihood solution it requires a careful initialization of the parameters and the optimal number of kernels in the mixture may be unknown beforehand. We propose a criterion based on the entropy of the pdf (probability density function) associated to each kernel to measure the quality of a given mixture model. Two different methods for estimating Shannon entropy are proposed and a modification of the classical EM algorithm to find the optimal number of kernels in the mixture is presented. We test our algorithm in probability density estimation, pattern recognition and color image segmentation.

Face recognition using probabilistic two-dimensional principal component analysis and its mixture model

Abstract: In this paper, by supposing a parametric Gaussian distribution over the image space (spanned by the row vectors of 2D image matrices) and a spherical Gaussian noise model for the image, we endow the two-dimensional principal component analysis (2DPCA) with a probabilistic framework called probabilistic 2DPCA (P2DPCA), which is robust to noise. Further, by using the probabilistic perspective of P2DPCA, we extend P2DPCA to a mixture of local P2DPCA models (MP2DPCA). MP2DPCA offers us a method of being able to model faces in unconstrained (complex) environment with possibly large variation. The model parameters could be fitted on the basis of maximum likelihood (ML) estimation via the expectation maximization (EM) algorithm. The experimental recognition results on UMIST face database confirm the effectivity of the proposed methods.

Color texture segmentation by decomposition of gaussian mixture model

Abstract: Recently we have proposed Gaussian mixtures as a local statistical model to synthesize artificial textures. We describe the statistical dependence of pixels of a movable window by multivariate Gaussian mixture of product components. The mixture components correspond to different variants of image patches as they appear in the window. In this sense they can be used to identify different segments of the source color texture image. The segmentation can be obtained by means of Bayes formula provided that a proper decomposition of the estimated Gaussian mixture into sub-mixtures is available. In this paper the mixture model is decomposed by maximizing the mean probability of correct classification of pixels into segments in a way taking into account the assumed consistency of final segmentation.

Dynamic events as mixtures of spatial and temporal features

Abstract: Dynamic events comprise of spatiotemporal atomic units. In this paper we model them using a mixture model. Events are represented using a framework based on the Mixture of Factor Analyzers (MFA) model. It is to be noted that our framework is generic and is applicable for any mixture modelling scheme. The MFA, used to demonstrate the novelty of our approach, clusters events into spatially coherent mixtures in a low dimensional space. Based the observations that, (i) events comprise of varying degrees of spatial and temporal characteristics, and (ii) the number of mixtures determines the composition of these features, a method that incorporates models with varying number of mixtures is proposed. For a given event, the relative importance of each model component is estimated, thereby choosing the appropriate feature composition. The capabilities of the proposed framework are demonstrated with an application: recognition of events such as hand gestures, activities.

On drag and lift forces in two-dimensional flows of a particulate mixture: A theoretical study

Abstract: In this paper we propose and derive expressions for the drag and lift forces in a two-phase particulate mixture. The analysis is limited to two-dimensional laminar flows. In the Section after the Introduction, a brief review of the single particle approach is provided; it is then shown that in most multiphase flow problems some generalization of these forces acting on a single particle is used. We then describe a different way of defining the lift force and the drag force, an approach used in non-Newtonian fluid mechanics. In the following Section, the essential equations of Mixture Theory are provided and the specific approach of [1] is used. In this scheme, the lift force is part of the interaction mechanisms, which are to be modeled as constitutive parameters. In the final Section, we derive an expression for the lift force, whereby it is shown that the normal component of the force acting on the body, obtained by integrating the traction vector of the mixture acting on a single isolated particle, will give us the desired expression for the lift force in multi-component flows.

Finite element simulation of deformation bands in saturated granular media with inhomogeneous porosities at the meso-scale

Abstract: The balance of mass and linear momentum of a solid-fluid mixture furnish a complete set of equations from which the displacements of the the solid matrix and the pore pressures can be resolved for the case of quasi-static loading, resulting in the so-called u — p Galerkin formulation. In this work, a recently proposed model for dense sands is utilized to model the effective stress response of the solid matrix appearing in the balance of linear momentum equation [1], [2], [3]. In contrast with other more traditional models, inherent inhomogeneities in the density field at the meso-scale can be easily incorporated and coupled with the macroscopic laws of mixture theory. The hydraulic conductivity is naturally treated as a function of the porosity in the solid matrix, hence allowing for a more realistic representation of the physical phenomenon. The aforementioned balance laws are cast into a fully nonlinear finite element program utilizing isoparametric elements satisfying the Babuska-Brezzi stability condition. Numerical simulations on dense sand specimens are performed to study the effects of inhomogeneities on the stability of saturated porous media at the structural level.

Numerical Approaches for microscopic modelling of solute redistribution during solidification of binary alloys

Abstract: In the present study, two numerical approaches for single-domain modelling of microsegregation during solidification of binary alloys are presented. In the first approach, the concentration jump at the moving solid/liquid interface is formulated using a volumetric term and a Boolean function. The governing solute redistribution equation, valid for the whole domain comprising the solid and liquid regions, is derived in terms of the liquid phase composition. The effects of microstructure coarsening on microsegregation has been described and included in the model. In the second approach, the continuum mixture theory is utilized to derive a single domain solute redistribution equation in terms of the mixture composition. The solidification front motion and dendrite arm coarsening effects are accommodated by considering the representative elementary volume to consist of solid, interdendritic, and extradendritic liquid phases. Numerical solutions have been obtained using a control-volume based finite-difference method with a fixed grid. Good agreement has been observed between the predictions of the present fixed-domain models and the exact analytical and experimental results.

Toward a New Paradigm for Reactive Flow Modeling

Abstract: Traditional reactive flow modeling provides a computational representation of shock initiation of energetic materials. Most reactive flow models require ad hoc assumptions to obtain robust simulations, assumptions that result from partitioning energy and volume change between constituents in a reactive mixture. For example, most models assume pressure and/or temperature equilibrium for the mixture. Many mechanical insults to energetic materials violate these approximations. Careful analysis is required to ensure that the model assumptions and limitations are not exceeded. One limitation is that the shock to detonation transition is replicated only for strong planar shocks. Many models require different parameters to match data from thin pulse, ramp wave, or multidimensional loading, an approach that fails for complex loading. To accurately simulate reaction under non‐planar shock impact scenarios a new formalism is required. The continuum mixture theory developed by Baer and Nunziato is used to eliminate ad hoc assumptions and limitations of current reactive flow models. This modeling paradigm represents the multiphase nature of reacting condensed/gas mixtures. Comparisons between simulations and data are presented.

Non orthogonal component analysis: application to anomaly detection

Abstract: Independent Component Analysis (ICA) has shown success in blind source separation. Its applications to remotely sensed images have been investigated recently. In this approach, a Linear Spectral Mixture (LSM) model is used to characterize spectral data. This model and the associated linear unmixing algorithms are based on the assumption that the spectrum for a given pixel in an image is a linear combination of the end-member spectra. The assumption that the abundances are mutually statistically independent random sources requires the separating matrix to be unitary. This paper considers a new approach, the Non Orthogonal Component Analysis (NOCA), which enables to relax this assumption. The experimental results demonstrate that the proposed NOCA provides a more effective technique for anomaly detection in hyperspectral imagery than the ICA approach. In particular, we highlight the fact that the difference between the performances of the two approaches increases when the number of bands decreases.

Concrete at Early Ages and Beyond: Numerical Model and Validation

Abstract: This work deals with a new mathematical/numerical model for the analysis of the behaviour of concrete considered as multiphase viscous porous material from early ages to long term periods. This is a solidification-type model where all changes of material properties are expressed as functions of hydration degree, and not maturity nor equivalent hydration period as in maturity-type models. A mechanistic approach has been used to obtain the governing equations, starting from micro-scale, by means of modified averaging theory, also called hybrid mixture theory, [1, 2]. Constitutive laws are directly introduced at macroscopic level. An evolution equation for the internal variable, hydration degree, describes hydration rate as a function of chemical affinity, considering additionally to the existing models, an effect of the relative humidity on the process. The model takes into account full coupling between hygral, thermal and chemical phenomena, as well as changes of concrete properties caused by hydration process, i.e. porosity, density, permeability, and strength properties. Phase changes and chemical phenomena, as well as the related heat and mass sources are considered. Some examples showing possibilities of the model for analysis of autogenous self-heating and selfdesiccation phenomena, as well as autogenous shrinkage are presented and discussed. Creep processes are modelled considering concrete as viscous-elastic material with aging caused by solidification of non-aging constituent, i.e. solidification theory for the so-called basic creep [1, 2]. A Kelvin-type chain has been chosen for the definition of the compliance function, which corresponds to an expansion of that function in a Dirichlet’s series. Shrinkage is defined using the effective stress principle, as usual in the mechanics of porous materials, and it is coupled to the creep model. In such a way it is possible to have creep strains even if the concrete structure is not externally loaded. Capillary shrinkage is, in fact, characterized from capillary tensions which can be seen as a sort of internal load for the microstructure of the material. A series of numerical computations compared to the experimental results are presented as validation of the model described above.

A Constitutive Damage Model for Dynamic Response of Dry/Water-Saturated Granite with Application to Study on Stress Waves

Abstract: A damage constitutive model of dry/water-saturated granite is proposed within the framework of continuum mechanics and mixture theory, and the model allows for the simulation of the effects of micro-cracks, micro-pores and saturated water. By implementing the model into wave propagation code, one dimensional stain waves in dry granite induced by a rectangular impulsive loading and spherical waves in water-saturated granite due to underground explosion are studied. The simulation results demonstrate the main features of the model, and it is shown that the theoretical model developed is valid for study on the characteristics of stress waves in rock medium.